A converse theorem for second-order modular forms of level N

Transcription

1 ACTA ARITHMETICA (2006 A converse theorem for second-order modular forms of level N by Özlem Imamoḡlu (Zürich and Yves Martin (Santiago 1. Introduction. The Eisenstein series twisted by modular symbols is the prime example of a second-order automorphic form. It was introduced by Goldfeld in [6] to study the distribution of modular symbols and to provide a new approach to Szpiro s conjecture. Since then, these series have been studied and generalized by many authors ([2], [4], [7], [8], [11] [13]. The twisted Eisenstein series is not an automorphic form in the classical sense but satisfies a shifted automorphy relation which involves the ordinary Eisenstein series. Automorphic functions of the same ind were also encountered by Kleban and Zagier in their research on crossing probabilities. In [9] they initiated the study of nth order modular forms, i.e. functions whose deviation from modularity is an (n 1th order modular form. In this paper we will restrict ourselves to the study of second order modular forms. Fix positive integers N and with even. Let H be the complex upper half-plane. If f = f(τ : H C is any function and γ = ( a b c d a matrix in GL + 2 (R, the usual slash operator is f(τ [γ] = (detγ /2 (cτ + d f ( aτ + b. cτ + d By linearity this defines an action of the group ring ZGL + 2 (R on the set of holomorphic functions on H. A second-order modular form F(τ of weight and level N is a function F : H C such that (i F(τ is holomorphic on H, (ii F(τ [γ] has at most polynomial growth at the cusps for each γ Γ 0 (N, 2000 Mathematics Subject Classification: Primary 11F66. Research of Y. Martin supported by Fondecyt grants , [361]

2 362 Ö. Imamoḡlu and Y. Martin (iii F(τ [(γ 1 I 2 (γ 2 I 2 ] = 0 for all γ 1, γ 2 in Γ 0 (N, (iv F(τ [γ I 2 ] = 0 for every parabolic element γ in Γ 0 (N. The C-vector space of second-order modular forms of weight and level N is denoted by M 2 (N. It has been investigated in [1] and [5]. In particular, a decomposition theorem in terms of elliptic modular forms is proved in [1], while in [5] the dimension of the space is found and a cohomological interpretation is given. Denote by M 1 (Γ the space of elliptic modular forms of weight over a group Γ SL 2 (R, and by M 1 (N the same space whenever Γ = Γ 0 (N. Clearly M 1 (N M2 (N. In most cases the containment is proper. By (i, (ii and (iv above, any second-order modular form has a Fourier series representation and hence an obvious Dirichlet series associated to it. In this note we investigate some properties of the latter. More precisely, for a Fourier series F(τ = n=0 A n exp(2πinτ and for s in C let ( 2π s L(F; s = A n n s and Λ N (F; s = Γ(sL(F; s. N n=1 For any primitive Dirichlet character χ of conductor m with gcd(n, m = 1 let ( 2π s L(F χ ; s = χ(na n n s and Λ Nm 2(F χ ; s = m Γ(sL(F χ ; s. N n=1 (The corresponding definition for the trivial character mod m is slightly different and is given in Section 4. Our main result is a converse theorem for M 2 (N. Theorem. Let {A n } n and {B n } n be two sequences in C with A n = O(n ν and B n = O(n ν for some ν > 0. Define functions F(τ = A n exp(2πinτ and G(τ = B n exp(2πinτ n=0 n=0 on H. Then statements (A and (B below are equivalent. (A F(τ and G(τ are in M 2 (N with G(τ = F(τ [w(n]. (B The following conditions on the corresponding Dirichlet series hold. The series Λ N (F; s + A 0 s + B 0 i s has a holomorphic continuation to the whole s-plane, is bounded on any vertical strip and satisfies Λ N (F; s = i Λ N (G; s.

3 Second-order modular forms 363 If a χ 0 (resp. bχ 0 is the constant term in the Fourier series of G χ (τ [w(np 2 ] (resp. F χ (τ [w(np 2 ], where χ is any Dirichlet character mod p and p is an odd prime with gcd(n, p = 1, then: (i The function i Λ Np 2(G χ ; s C χ Λ Np 2(F χ ; s + aχ 0 s i C χ b χ 0 s has a holomorphic continuation to the whole s-plane, bounded on any vertical strip. (ii The function i Λ Np 2(G χ ; s C χ Λ Np 2(F χ ; s is the (completed L-function of a modular form in M 1 (Γ 0(Np 2 Γ 1 (p. (iii The Dirichlet series χ χ(uc χw(χλ Np 2(G χ ; s (where the sum is over all Dirichlet characters mod p has a holomorphic continuation to the whole s-plane, is bounded on vertical strips and satisfies χ(uc χ W(χ{i Λ Np 2(G χ ; s C χ Λ Np 2(F χ ; s} = 0, χ whenever u Z is such that (p ± (modun. Here and from now on, w(x = ( 0 1 x 0 in GL + 2 (R, W(χ is the Gauss sum associated to χ and C χ = χ( NW(χW(χ 1. Weil s converse theorem allows us to replace condition (ii above by a set of functional equations. In Section 5 we rephrase our theorem in those terms (see Theorem 2. After the completion of this wor we learned about a preprint of Diamantis, Knopp, Mason and O Sullivan [3]. They study a similar Dirichlet series and prove a partial converse theorem (Thm. 14 in [3]. Their result differs from ours in two ways. First, the space of forms considered in [3] is not the one studied in this paper. Secondly, in the converse direction they impose a strong periodicity relation which we do not assume here. This article is organized as follows. In the next section we recall Weil s converse theorem in the form that we need. In Section 3 we give the definition of a second-order modular form. Then, in the following section we give an equivalent description of those second-order modular forms using their Fourier series and Dirichlet characters. Finally, in Section 5 we state and prove the converse theorem of the title. Notation. For convenience we use the symbols I 2 and θ(x for, respectively, the matrices ( 1 0 ( and 1 x in GL + 2 (R. Also, we write e(x for exp(2πix.

4 364 Ö. Imamoḡlu and Y. Martin 2. Weil s converse theorem. In this section we recall Weil s converse theorem and some standard lemmas which we will need. For their proofs see for example [10, p. 117]. Lemma 1. Let f : H C be a holomorphic function with a Fourier series representation f(τ = n=0 a ne(nτ which converges absolutely and uniformly on compact subsets of H. If there is a real number ν > 0 such that f(τ = O(Im(τ ν as Im(τ 0 uniformly in Re(τ, then a n = O(n ν. Lemma 2. Let {a n } n=0 be a sequence of complex numbers and set (1 f(τ = a n e(nτ for τ H. n=0 If a n = O(n ν for some real number ν > 0, then the series (1 converges absolutely and uniformly on compact subsets of H. In particular f is a holomorphic function on H. Moreover both f(τ = O(Im(τ ν 1 as Im(τ 0 and f(τ a 0 = O(e 2π Im(τ as Im(τ, uniformly in Re(τ. By these results, the holomorphic functions on H satisfying the hypothesis of Lemma 1 correspond bijectively to the sequences {a n } n in C with a n = O(n ν for some ν > 0. Definition 1. Let f(τ = n=0 a ne(nτ be a function as in Lemma 1. For s in C define ( 2π s L(f; s = a n n s and Λ N (f; s = Γ(sL(f; s. N n=1 Let χ be a primitive Dirichlet character of conductor m with gcd(n, m = 1. For s in C define ( 2π s L(f, χ; s = χ(na n n s and Λ Nm 2(f, χ; s = m Γ(sL(f, χ; s. N n=1 Since a n = O(n ν for some ν > 0, the series L(f; s, L(f, χ; s, Λ N (f; s and Λ Nm 2(f, χ; s are absolutely and uniformly convergent on compact subsets of the half-plane Re(s > ν + 1. The following result is due to Hece. Proposition 1. Let f(τ = n=0 a ne(nτ and g(τ = n=0 b ne(nτ be functions on H satisfying the hypothesis of Lemma 1. Then statements (A and (B below are equivalent. (A g(τ = f(τ [w(n]. (B Λ N (f; s and Λ N (g; s admit meromorphic continuations to the whole s-plane and they satisfy Λ N (f; s = i Λ N (g; s.

5 Second-order modular forms 365 Moreover Λ N (f; s+a 0 /s+i b 0 /( s is holomorphic on the whole s-plane and bounded on any vertical strip. Hece used Proposition 1 to get his converse theorem for elliptic modular forms over SL 2 (Z. André Weil [15] proved such a converse theorem for modular forms over congruence subgroups Γ 0 (N by considering Fourier series twisted with Dirichlet characters. Definition 2. Let f : H C be a holomorphic function represented by a Fourier series f(τ = n=0 a ne(nτ, and χ any primitive Dirichlet character of conductor m. The Fourier series twisted by χ is f χ (τ = n=0 χ(na ne(nτ. Notice that Λ Nm 2(f χ ; s = Λ Nm 2(f, χ; s. A standard computation yields (2 χ(uf(τ [θ(u/m] = W(χf χ (τ. u(mod m Next, we recall Weil s converse theorem in a form that is convenient for our purposes. Let m be a positive integer with gcd(n, m = 1. We denote by M(N, m any set of odd prime numbers relatively prime to Nm 2 and congruent to 1 mod m, such that M(N, m {a + lcnm 2 l Z} is not empty whenever a and c are integers with gcd(cnm 2, a = 1 and a 1 (modm. (Such a set always exists by Dirichlet s theorem on primes in arithmetic progressions. Theorem 1. Let {a n } n and {b n } n be two sequences in C with a n = O(n ν and b n = O(n ν for some ν > 0. Define functions f(τ = a n e(nτ and g(τ = b n e(nτ n=0 n=0 on H. Then statements (A and (B below are equivalent: (A f(τ and g(τ are in M 1 (Γ 0(Nm 2 Γ 1 (m and satisfy g(τ = f(τ [w(nm 2 ]. (B The following conditions on the corresponding Dirichlet series hold. The series Λ Nm 2(f; s+a 0 /s+i b 0 /( s has a holomorphic continuation to the whole s-plane, is bounded on any vertical strip and satisfies (3 Λ Nm 2(f; s = i Λ Nm 2(g; s. For any primitive Dirichlet character ψ whose conductor t is in M(N, m, the series Λ Nm 2 t2(f, ψ; s has a holomorphic continuation to the whole s-plane, is bounded on any vertical strip and satisfies (4 Λ Nm 2 t 2(f, ψ; s = i ψ(m 2 C ψ Λ Nm 2 t2(g, ψ; s.

6 366 Ö. Imamoḡlu and Y. Martin 3. Definitions and basic properties. Recall that the formal definition of second-order modular forms of level N was given in the introduction. In the first result of this section we describe them in a slightly different way. Lemma 3. A function F : H C is in M 2 (N if and only if (a F(τ is holomorphic on H, (b F(τ has at most polynomial growth at the cusps, (c F(τ [γ I 2 ] M 1 (N for any γ = ( p v un q in Γ0 (N such that u = 0 or whose entries p, q are two distinct odd primes, (d F(τ [γ I 2 ] = 0 for every matrix γ as above such that u = 0 or p + q ±2 (modun. Proof. Clearly the definition of second-order modular forms yields (a, (b and (c. In fact, (c holds for any matrix γ in Γ 0 (N. As for (d, we observe that (iv implies (5 F(τ ( ] 1 l [± = F(τ for all integers l. If γ = ( p v un q is in Γ0 (N with u 0 and p + q ±2 (modun, we can write p + q = ±2 + unl for some l in Z. Then ( ( 1 l p γ = q unl is a parabolic element of Γ 0 (N. Consequently, (iii and (iv yield F(τ [γ I 2 ] = F(τ [γ I 2 ] [( ] 1 l = F(τ [γ I 2 ] [( ] 1 l + F(τ [( ] 1 l I 2 = F(τ ( ] 1 l [γ I 2 = 0. Next we prove the equivalence in the other direction. If γ = ( a b cn d Γ0 (N with c 0 we consider the integral sequences {a + lcn l Z} and {d + lcn l Z}. By Dirichlet s theorem on primes in arithmetic progressions, each of them contains infinitely many primes. In particular, there are arbitrarily large odd primes p q of the form p = a + tcn, q = d + scn for some integers t, s. Put u = c and v = (b + sp + stun + qt. Then ( ( ( p v 1 t 1 s γ = Γ 0 (N and γ = γ. un q

7 Second-order modular forms 367 Consequently, from (5 and hypothesis (c one gets (6 F(τ [γ I 2 ] = F(τ [( ] [ ( ] 1 t 1 s I 2 γ + F(τ [ ( ] 1 s γ I 2 = F(τ [ γ I 2 ] [( ] 1 s + F(τ [( ] 1 s I 2 = F(τ [ γ I 2 ]. Thus F(τ [γ I 2 ] is in M 1 (N and (iii follows. Suppose that the matrix γ above is also parabolic. Then a + d = ±2. If c = 0 there is nothing to prove. Otherwise p + q = a + d + (t + scn ±2 (modun. Hence (6 and (d yield (iv. We have already shown that F(τ [γ I 2 ] M 1 (N for all γ Γ 0(N. Hence these functions must have polynomial growth at the cusps. As the same is true for F(τ, we obtain (ii. Notice that our proof of (d from the definition of second-order modular forms is valid for any matrix γ = ( a b cn d in Γ0 (N with a+d ±2 (modcn. Consider next the following generalization of condition (d: For a holomorphic function F : H C, let (D F(τ [γ I 2 ] = 0 for every matrix γ = ( p v un q in Γ0 (N such that u = 0 or p + q ±2 (modun with p an odd prime. From equation (6 with s = 0 one can show that (D implies part (iv in the definition of a second-order modular form. 4. Fourier series and Dirichlet characters. In this section we give a characterization of a second-order modular form using its Fourier series at infinity and twists of it by Dirichlet characters. Throughout this section F(τ and G(τ are always holomorphic functions on H represented by the Fourier series F(τ = n=0 A ne(nτ and G(τ = n=0 B ne(nτ with A n, B n in C, A n = O(n ν and B n = O(n ν for some ν > 0. Moreover they satisfy G(τ = F(τ [w(n]. Any γ in Γ 0 (N defines functions f γ, g γ : H C by f γ (τ = F(τ [γ I 2 ] and g γ (τ = G(τ [γ I 2 ]. For convenience we write f p,v (τ (resp. g p,v (τ instead of f γ (τ (resp. g γ (τ if γ = ( p v un q Γ0 (N. A straightforward computation shows that the notation is not ambiguous. Let f : H C be a holomorphic function represented by the series f(τ= n=0 a ne(nτ. We have already defined f χ (τ for a primitive Dirichlet

8 368 Ö. Imamoḡlu and Y. Martin character χ. It is also convenient to define f χ0 (τ for the trivial character χ 0 mod m whenever m is a prime. Set f χ0 (τ = f(τ m m n a ne(nτ. One reason for such a definition is that f χ0 (τ satisfies an identity lie (2. Namely (7 a n e(nτ. u (mod mf(τ [θ(u/m] = m m n The analogy is more evident once we observe that W(χ 0 = 1. Lemma 4. Let m be a positive integer with gcd(n, m = 1. If χ is either a primitive Dirichlet character of conductor m or the trivial Dirichlet character mod m with m prime, then (8 G χ (τ [w(nm 2 ] C χ F χ (τ = C χ W(χ 1 v (mod m χ(vf m,v (τ [θ(v/m]. Proof. Let u, v be integers such that uvn 1 (modm. Then (9 θ(u/mw(nm 2 θ( v/m = mw(nγ m,v for some γ m,v = ( m v un in Γ0 (N. If χ is a primitive (resp. trivial character, we use (2 (resp. (7 and (9 to get G χ (τ [w(nm 2 ] = W(χ 1 χ(uf(τ [γ m,v θ(v/m] = W(χ 1 χ( N u(mod m v (mod m = C χ F χ (τ + C χ W(χ 1 χ(v{f(τ + f m,v (τ} [θ(v/m] v (mod m χ(vf m,v (τ [θ(v/m]. Notice that f p,v (τ is not defined if gcd(v, m > 1, but then χ(v = 0. Definition 3. Let m > 1 be an integer relatively prime to N. If χ is any primitive Dirichlet character of conductor m or the trivial Dirichlet character mod m with m prime, let H χ (τ be the left hand side of (8, i.e. (10 H χ (τ = G χ (τ [w(nm 2 ] C χ F χ (τ. Let p be a prime number with gcd(n, p = 1. Then every Dirichlet character mod p is either primitive or trivial. Hence equation (8 and the orthogonality relations for characters yield (11 (p 1f p,v (τ [θ(v/p] = χ χ(vc χ W(χH χ (τ for any integer v with gcd(v, p = 1. Here the sum is over all characters mod p.

9 Second-order modular forms 369 Next we recall some standard notation and prove a technical lemma. Let M and m be two positive integers. As usual, the symbols Γ0 0 (M, m and Γ 1 (m denote the groups {( } Γ0 0 a b (M, m = SL 2 (Z c d c 0 (modm, b 0 (modm and {( } Γ 1 a b (m = SL 2 (Z c d a 1 (modm, b 0 (modm. Lemma 5. Let p and q be two distinct odd primes such that pq uvn = 1 for some u, v in Z. Then Γ 0 (Np 2 Γ 1 (p, Γ0 0(N, q2 Γ 1 (q (the subgroup of SL 2 (Z generated by the two intersections is equal to Γ 0 (N. Proof. Let ( a b cn d Γ0 (N with a 1 (modq. There are integers x, y such that ( x yq 2 Γ0 0 (N, q 2 Γ 1 (q. cn a Thus ( ( x yq 2 a cn a cn b = d ( is in Γ 0 (Np 2 Γ 1 (p as desired. Let ( a b cn d Γ0 (N with c relatively prime to q. There exists c Z such that cc 1 (modq. Therefore ( ( ( 1 uvc (a 1 a b a + uvc (a 1cN = Γ 0 (N cn d cn d with a + uvc (a 1cN 1 (modq. By the previous argument this matrix is in Γ 0 (Np 2 Γ 1 (p, Γ0 0(N, q2 Γ 1 (q, and hence so is ( a b cn d. Finally, we loo at the case ( a b cn d Γ0 (N with c 0 (modq. As p q there is a Z such that aa 1 (modq and a 1 (modpn. Hence there are integers d, b such that ( a b Np 2 d Γ 0 (Np 2 Γ 1 (p and ( ( a b a Np 2 d cn ( b a a + b cn = d Np 2 a + d cn Γ 0 (N with a a + b cn 1 (modq. By the previous argument this matrix, and therefore ( a b cn d, is in Γ0 (Np 2 Γ 1 (p, Γ0 0(N, q2 Γ 1 (q.

10 370 Ö. Imamoḡlu and Y. Martin The next proposition describes a second-order modular form in terms of the functions H χ (τ given in (10. It is a ey result for the proof of our main theorem. Proposition 2. F(τ is in M 2 (N if and only if the following four statements hold: (a F(τ is holomorphic on H. (b F(τ has at most polynomial growth at the cusps. (c For any odd prime p relatively prime to N and arbitrary Dirichlet character χ mod p, H χ (τ M 1 (Γ 0(Np 2 Γ 1 (p. (d For any odd prime p relatively prime to N and arbitrary u Z with (p ± (modun, χ(uc χ W(χH χ (τ = 0 χ where the sum is over all Dirichlet characters χ mod p. Proof. First we assume that F(τ is in M 2 (N. For any integer v relatively prime to p consider the matrix γ p,v in Γ 0 (N determined by equation (9. Then f p,v (τ = F(τ [γ p,v I 2 ] is in M 1 (N (see proof of Lemma 3 and therefore f p,v (τ [θ(v/p] is in M 1 (Γ 0(Np 2 Γ 1 (p. This fact and Lemma 4 yield (c. In order to deduce (d we write (p ± 1 2 = lun for some l in Z. Then lun 1 (modp and therefore l v (modp for any integer v such that uvn 1 (modp. Put l = v + tp for some t in Z. Clearly (p ± 1 2 = (tp vun, i.e. p vun = 2 + tun. p Notice that q = (1 + vun/p Z, ( p v un q Γ0 (N and p + q ±2 (modun. By Lemma 3 we conclude that f p,v (τ [θ(v/p] = 0. Now we use (11 and the congruence uvn 1 (modp to get (d. Next we prove the converse implication. Evidently, it suffices to show that (a, (b, (c and (d imply statements (c and (d of Lemma 3. Let ( p v 0 γ p,v0 = Γ 0 (N u 0 N q with p, q two distinct odd primes. Then gcd(n, p = gcd(n, q = 1 and gcd(v 0, p = 1 (resp. gcd(u 0, q = 1. From (c and (11 we infer that f p,v0 (τ [θ(v 0 /p], and hence f p,v0 (τ, are in M 1 (Γ 0(Np 2 Γ 1 (p.

11 Second-order modular forms 371 On the other hand, w(np 2 is in the normalizer of the group Γ 0 (Np 2 Γ 1 (p, w(np 2 2 = Np 2 I 2 and C χ C χ = 1. Hence (c also implies H χ (τ [w(np 2 ] M 1 (Γ 0(Np 2 Γ 1 (p for all Dirichlet characters χ mod p. Using the previous argument with q (resp. G(τ instead of p (resp. F(τ we deduce that g q, u0 (τ M 1 (Γ 0(Nq 2 Γ 1 (q. The identities f p,v0 (τ [w(n] = g q, u0 (τ and w(n 1 (Γ 0 (Nq 2 Γ 1 (qw(n = Γ 0 0 (N, q 2 Γ 1 (q are easy to chec. Thus f p,v0 (τ M 1 ( Γ 0(Np 2 Γ 1 (p, Γ 0 0 (N, q 2 Γ 1 (q. By Lemma 5 we conclude that f p,v0 (τ is in M 1 (N, which is (c. Finally, consider a matrix ( p v un q in Γ0 (N whose entries p, q are distinct odd primes such that p+q ±2 (modun. Since pq 1 (modun we have p(±2 p 1 (modun and therefore (p (modun. By hypothesis (d, equation (11 and the relation uvn 1 (modp, we get f p,v (τ [θ(v/p] = 0 and therefore f p,v (τ = The main theorem. In this section we first state an auxiliary proposition and introduce a new ind of twist of Fourier series with Dirichlet characters. Then we restate our main result in terms of different twists of Dirichlet series and give the proof of it. Proposition 3. Let F 1 (τ = n=0 A ne(nτ, f 1 (τ = n=0 a ne(nτ, F 2 (τ = n=0 B ne(nτ and f 2 (τ = n=0 b ne(nτ be functions on H satisfying the hypothesis of Lemma 1. Let M be a positive integer. Then statements (A and (B below are equivalent. (A F 1 (τ + f 1 (τ = F 2 (τ [ω(m] and f 1 (τ = f 2 (τ [ω(m]. (B Λ M (F 1 ; s and Λ M (F 2 ; s admit meromorphic continuations to the whole s-plane and they satisfy and Moreover, both i Λ M (F 2 ; s = Λ M (F 1 ; s + Λ M (f 1 ; s Λ M (f 2 ; s = i Λ M (f 1 ; s. Λ M (F 1 ; s + A 0 s + B i 0 s b 0 i s and Λ M(f 1 ; s + a 0 s + b 0 i s are holomorphic on the whole complex s-plane and bounded on any vertical strip. Proof. We can argue exactly as in the proof of Proposition 1. (See for example [10, pp ].

12 372 Ö. Imamoḡlu and Y. Martin Definition 4. Let f(τ = n=0 a ne(nτ and g(τ = n=0 b ne(nτ be holomorphic functions on H satisfying the hypothesis of Lemma 1 and the relation f(τ [w(n] = g(τ. Let m be a positive integer with gcd(n, m = 1 and χ a primitive Dirichlet character of conductor m or the trivial character mod m with m prime. We define f χ (τ = g χ (τ [w(nm 2 ]. The function f χ (τ is similar to the twist of f(τ by χ. Compare for example the description of f χ (τ in (2 with the trivial identity (12 f χ (τ = W(χ 1 χ(uf(τ [w(nθ(u/mw(nm 2 ]. u(mod m In the particular case that f(τ is a modular form in M 1 (N one has fχ (τ = C χ f χ (τ. It is easy to see that the definition above and a straightforward application of Theorem 1 allow us to rephrase our main result (the Theorem in the introduction as Theorem 2 below. In the latter form the similarities with Weil s converse theorem are more evident. Theorem 2. Let {A n } n and {B n } n be two sequences in C with A n = O(n ν and B n = O(n ν for some ν > 0. Define functions F(τ = A n e(nτ and G(τ = B n e(nτ n=0 n=0 on H. Then statements (A and (B below are equivalent. (A F(τ and G(τ are in M 2 (N with G(τ = F(τ [w(n]. (B The following conditions on the corresponding Dirichlet series hold. The series Λ N (F; s + A 0 /s + i B 0 /( s has a holomorphic continuation to the whole s-plane, is bounded on any vertical strip and satisfies (13 Λ N (F; s = i Λ N (G; s. If a χ 0 (resp. bχ 0 is the constant term in the Fourier series of F χ (τ (resp. G χ (τ, where χ is any Dirichlet character mod p and p is an odd prime with gcd(n, p = 1, then: (B.1 The function Λ Np 2(F χ ; s C χ Λ Np 2(F χ ; s + aχ 0 s b χ i 0 C χ s has a holomorphic continuation to the whole s-plane bounded on any vertical strip. (B.2 For any primitive Dirichlet character ψ of conductor t in M(N, p, the function Λ Np 2 t 2(F χ, ψ; s C χ Λ Np 2 t 2(F χ, ψ; s

13 Second-order modular forms 373 has a holomorphic continuation to the whole s-plane, and is bounded on any vertical strip. Moreover (14 Λ Np 2 t 2(F χ, ψ; s C χ Λ Np 2 t 2(F χ, ψ; s (15 = i ψ(p 2 C ψ Λ Np 2 t 2(G χ, ψ; s i ψ(p 2 C ψ C χ Λ Np 2 t 2(Gχ, ψ; s. (B.3 The Dirichlet series χ χ(uc χw(χλ Np 2(G χ ; s (where the sum is over all Dirichlet characters mod p has a holomorphic continuation to the whole s-plane, is bounded on vertical strips and satisfies χ(uc χ W(χ{i Λ Np 2(G χ ; s C χ Λ Np 2(F χ ; s} = 0 χ whenever u Z is such that (p ± (modun. Proof. Assume that F(τ is in M 2 (N and G(τ = F(τ [w(n]. Then F(τ and G(τ satisfy the hypothesis of Lemma 1 and by Proposition 3 we get the first part of (B, i.e. the function Λ N (F; s+a 0 /s+i B 0 /( s has a holomorphic continuation to the whole s-plane, is bounded on any vertical strip and satisfies (13. Now let χ be a character as in the theorem. Proposition 2 shows that H χ (τ = F χ (τ C χ F χ (τ is in M 1 (Γ 0(Np 2 Γ 1 (p. Hence Theorem 1 implies that Λ Np 2(F χ ; s C χ Λ Np 2(F χ ; s + a χ 0 /s i C χ b χ 0 /( s has holomorphic continuation to the whole s-plane bounded on any vertical strip. We also use Theorem 1 to conclude that for any primitive Dirichlet character ψ as above the series Λ Np 2 t 2(F χ C χ F χ, ψ; s has a holomorphic continuation to the whole s-plane, is bounded on any vertical strip and satisfies Λ Np 2 t 2(F χ C χ F χ, ψ; s = i ψ(p 2 C ψ Λ Np 2 t 2(G χ C χ G χ, ψ; s. From these equations and the linearity of the L-function we get (14. For the proof of (B.3 consider F p,u (τ = χ χ(uc χ W(χG χ (τ and G p,u (τ = χ χ(uw(χf χ (τ, where the sums are over all Dirichlet characters mod p and u is any integer such that (p ±1 2 0 (modun. Since the functions defined by the Fourier series F χ (τ and G χ (τ satisfy the hypothesis of Lemma 1, the same is true for F p,u (τ and G p,u (τ.

14 374 Ö. Imamoḡlu and Y. Martin Part (d of Proposition 2 is equivalent to F p,u (τ [w(np 2 ] = G p,u (τ, and this identity yields (B.3 by Proposition 1. For the converse implication we argue as follows. The growth of the Fourier coefficients A n, B n, together with Lemma 2, implies that both series F(τ and G(τ define holomorphic functions on H. From the particular meromorphic continuation of Λ N (F; s to the s-plane, equation (13 and Proposition 1 we get G(τ = F(τ [w(n]. Next we consider any odd prime p with gcd(n, p = 1 and any Dirichlet character χ mod p. Claim. The function H χ (τ is invariant under the translation τ τ+1. Proof of the Claim. Let F p,u (τ and G p,u (τ be as above. We use Proposition 1 to deduce from (B.3 that (16 G p,u (τ = F p,u (τ [w(np 2 ]. Notice that (16 is equivalent to the equation in part (d of Proposition 2. Arguing as in the proof of the latter we can deduce from (16 the invariance of F(τ under ( p v un q in Γ0 (N where q is any integer with p + q ±2 (modun. Now we recall the remar after Lemma 3 and conclude that F(τ is invariant under any parabolic element of Γ 0 (N. Since the conjugate of any parabolic matrix by w(n is parabolic, G(τ = F(τ [w(n] is also invariant under any parabolic element of Γ 0 (N. This fact and the identities ( ( w(np = Np 2 w(np 2 1 and θ ( ( u 1 0 p Np 2 1 = ( 1 unp u 2 N Np unp θ ( u p imply that F χ (τ = G χ (τ [w(np 2 ] is invariant under the translation τ τ + 1. This implies the Claim. Consequently, H χ (τ = n Z a ne(nτ for some a n C. Using now the estimate G χ (τ = O(Im(τ ν 1 as Im(τ 0 we conclude that H χ (τ = n 0 a ne(nτ. Similarly one has H χ (τ [w(np 2 ] = n 0 b ne(nτ for some coefficients b n in C. Proposition 3 then implies Λ Np 2(H χ ; s = i Λ Np 2(G χ ; s C χ Λ Np 2(F χ ; s. Notice that the linearity of the L-function yields

15 Second-order modular forms 375 Λ Np 2 t 2(H χ, ψ; s = Λ Np 2 t 2(F χ, ψ; s C χ Λ Np 2 t 2(F χ, ψ; s and Λ Np 2 t 2(H χ [w(np 2 ], ψ; s = Λ Np 2 t 2(G χ, ψ; s C χ Λ Np 2 t 2(Gχ, ψ; s. If we now use the hypothesis in (B.2 and Theorem 1 we conclude that H χ (τ is in M 1 (Γ 0(Np 2 Γ 1 (p. We already now that F(τ [γ I 2 ] = 0 for any parabolic γ in Γ 0 (N. This identity and the estimate F(τ = O(Im(τ ν 1 as Im(τ 0 (see Lemma 2 imply that F(τ has at most polynomial growth at every cusp of Γ 0 (N (details as in [10, p. 41] for example. By Proposition 2 we conclude that F(τ is in M 2 (N. Finally, we want to point out that it is possible to refine our converse Theorem 2 to a statement where only finitely many twists of F(τ and G(τ are necessary. One can proceed as in [14] where Razar maes such a refinement of Weil s theorem. References [1] G. Chinta, N. Diamantis and C. O Sullivan, Second order modular forms, Acta Arith. 103 (2002, [2] G. Chinta and D. Goldfeld, Grössencharater L-functions of real quadratic fields twisted by modular symbols, Invent. Math. 144 (2001, [3] N. Diamantis, M. Knopp, G. Mason and C. O Sullivan, L-functions of second order cusp forms, preprint. [4] N. Diamantis and C. O Sullivan, Hece theory of series formed with modular symbols and relations among convolution L-functions, Math. Ann. 318 (2000, [5],, The dimensions of spaces of holomorphic second order automorphic forms and their cohomology, preprint, arxiv:math.nt/ [6] D. Goldfeld, Zeta functions formed with modular symbols, in: Proc. Sympos. Pure Math. 66, Amer. Math. Soc., 1999, [7] D. Goldfeld and E. Gunnells, Eisenstein series twisted by modular symbols for the group GL n, Math. Res. Lett. 7 (2000, [8] J. Jorgenson and C. O Sullivan, Convolution Dirichlet series and a Kronecer limit formula for second-order Eisenstein series, Nagoya Math. J. 179 (2005, [9] P. Kleban and D. Zagier, Crossing probabilities and modular forms, J. Statist. Phys. 113 (2003, [10] T. Miyae, Modular Forms, Springer, Berlin, [11] C. O Sullivan, Properties of Eisenstein series with modular symbols, J. Reine Angew. Math. 518 (2000, [12] Y. Petridis, Spectral deformations and Eisenstein series associated with modular symbols, Int. Math. Res. Not. 2002, no. 19, [13] Y. Petridis and M. S. Risager, Modular symbols have a normal distribution, Geom. Funct. Anal. 14 (2004,

16 376 Ö. Imamoḡlu and Y. Martin [14] M. Razar, Modular forms for Γ 0 (N and Dirichlet series, Trans. Amer. Math. Soc. 231 (1997, [15] A. Weil, Über die Bestimmung Dirichletscher Reihen durch Funtionalgleichungen, Math. Ann. 168 (1967, Department of Mathematics ETH Zürich CH-8096 Zürich, Switzerland ozlem@math.ethz.ch Departamento de Matemáticas Facultad de Ciencias Universidad de Chile Casilla 653, Santiago, Chile ymartin@uchile.cl Received on and in revised form on (4995